arXiv subjects
Publications and source records attributed to A. A. Buturlakin.
For a set of primes $π$, denote by $E_π$ the class of finite groups containing a Hall $π$-subgroup. We establish that $E_{π_1}\cap E_{π_2}$ is contained in $E_{π_1\capπ_2}$. As a corollary, we prove that if $π$ is a set of primes, $l$ is an integer such that $2\leqslant l<|π|$ and $G$ is a finite group that contains a Hall $ρ$-subgroup for every subset $ρ$ of $π$ of size $l$, then $G$ contains a solvable Hall $π$-subgroup.
Consider a triangle $ABC$ with given lengths $l_a,l_b,l_c$ of its internal angle bisectors. We prove that in general, it is impossible to construct a square of the same area as $ABC$ using a ruler and compass. Moreover, it is impossible to express the area of $ABC$ in radicals of $l_a,l_b,l_c$.
Let $G$ be a finite group and let $π$ be a set of primes. In this paper, we prove a criterion for the existence of a solvable $π$-Hall subgroup of $G$, precisely, the group $G$ has a solvable $π$-Hall subgroup if, and only if, $G$ has a $\{p,q\}$-Hall subgroup for any pair $p$, $q\inπ$.
The supremum of lengths of strict chains of nested centralizers is called the $c$-dimension (centralizer dimension) of $G$. We prove two structure theorems for locally finite groups of finite $c$-dimension. We also prove that the $c$-dimension of the quotient of such a group by a locally soluble radical is bounded in terms of the $c$-dimension of the group itself.
The spectra of a finite group is the set of its element orders. We obtain an arithmetic description of finite symplectic and orthogonal groups. In particular, a description of spectra of all finite simple simplectic and orthogonal groups is obtained.